Fraction Calculator

A fraction calculator earns its place when it does more than print an answer. This one adds, subtracts, multiplies and divides two fractions, mixed numbers or decimals, reduces the result to lowest terms, and then shows the same answer three ways at once: as a simplified fraction, as a mixed number and as a decimal. Under the answer it prints the working, one line per step, in the order a teacher expects to see it. If you are checking homework, that is the part you actually need, because the answer alone does not tell you where your own attempt went wrong.

How to enter fractions, mixed numbers and decimals

Each fraction has three boxes: a whole number, a numerator and a denominator. A plain fraction such as 3/4 uses the numerator and denominator and leaves the whole number blank. A mixed number such as 1 3/4 uses all three. A whole number such as 7 goes in the first box alone. Decimals are accepted in any box, so typing 0.125 in the numerator with a denominator of 1 turns the calculator into a decimal to fraction converter without a separate mode. Those five input shapes, 3/4, 1 3/4, -1 3/4, 7 and 0.125, are the ones a fraction tool is expected to understand.

The minus sign deserves a sentence of its own. A minus written in front of a mixed number applies to the whole quantity, not just to the whole number part. So -1 3/4 means minus one and three quarters, which is -7/4. It does not mean -1 plus 3/4, which would be -1/4. Put the minus on the whole number box and the calculator treats the entire mixed number as negative, and the working line says so.

Adding and subtracting: finding a common denominator

Two fractions can only be added once they count the same size of piece, which means they need the same denominator. The general rule is a/b + c/d = (ad + bc)/bd, and for subtraction a/b - c/d = (ad - bc)/bd. Take 3/4 + 1/6. Multiplying the denominators gives 24, so the sum is 18/24 + 4/24 = 22/24, which reduces to 11/12. Subtraction runs the same way: 3/4 - 1/6 = 18/24 - 4/24 = 14/24, or 7/12.

Multiplying the denominators always works. What it costs you is a bigger number to reduce at the end. The least common denominator, or LCD, is the smallest number both denominators divide into, and using it keeps the numbers small. For 3/4 + 1/6 the LCD is 12, so the sum is 9/12 + 2/12 = 11/12 with nothing left to simplify. Neither route is more correct than the other, and both land on the same fraction in lowest terms. The LCD is a convenience, not a requirement.

The quick way to find the LCD is to divide the product of the denominators by their greatest common factor: LCD = b × d ÷ GCF(b, d). For 4 and 6 that is 24 ÷ 2 = 12. There is also a shortcut worth spotting. When the smaller denominator divides the larger one exactly, the larger one is already the LCD, so for 1/4 + 3/8 you work in eighths straight away. When the denominators share no factor at all, as with 1/3 + 1/5, the LCD is simply their product: 5/15 + 3/15 = 8/15. The calculator picks the right case and names it in the working.

Mixed numbers are converted to improper fractions first. 1 1/4 + 1 1/3 becomes 5/4 + 4/3, then 15/12 + 16/12, then 31/12, which is 2 7/12. Negative fractions follow the ordinary sign rules: -3/4 + -5/8 is -6/8 - 5/8 = -11/8, or -1 3/8.

Multiplying fractions

Multiplication needs no common denominator. Multiply the numerators together and the denominators together, a/b × c/d = ac/bd, then reduce. 2/3 × 5/6 = 10/18, which reduces to 5/9. 3/4 × 1/6 = 3/24, which reduces to 1/8. The word "of" in a recipe or a word problem means multiply, so two thirds of five sixths is the same 5/9. A product of two proper fractions is always smaller than either of them, which is a useful sanity check on any answer.

Dividing fractions: keep, change, flip

To divide by a fraction, multiply by its reciprocal. Keep the first fraction, change the division sign to multiplication, and flip the second fraction upside down. In symbols, a/b ÷ c/d = a/b × d/c = ad/bc. So 3/4 ÷ 1/6 becomes 3/4 × 6/1 = 18/4, which reduces to 9/2, or 4 1/2. Likewise 1/2 ÷ 3/5 = 5/6 and 2/5 ÷ 3/7 = 14/15.

The flip is not a trick. Division asks what you multiply the second number by to get the first. The standard school example is 2/3 ÷ 3/4 = 8/9, and you can check it by going backwards: 3/4 of 8/9 is 24/36, which is 2/3. The same logic explains the one case the calculator refuses. Dividing by 0/5 is dividing by zero, because 0/5 is worth nothing, and no number multiplied by zero can give you back a fraction that is not zero. A zero in the numerator of the second fraction is just as undefined as a zero in a denominator, even though it looks harmless.

Simplifying fractions to lowest terms

A fraction is in lowest terms, also called simplest form, when the numerator and denominator share no factor bigger than 1. To get there, find their greatest common factor and divide both by it. 4/8 has a GCF of 4, giving 1/2. 8/12 has a GCF of 4, giving 2/3. 45/10 has a GCF of 5, giving 9/2. The familiar reductions 6/9 = 2/3, 4/12 = 1/3, 14/16 = 7/8 and 15/24 = 5/8 all come from the same move, and a negative fraction reduces exactly like its positive twin.

You can find the GCF by prime factorisation. For 117/52, 117 = 3 × 3 × 13 and 52 = 2 × 2 × 13, the shared factor is 13, and the fraction becomes 9/4. Once the numbers pass two digits, though, the Euclidean algorithm is faster by hand. Divide the larger number by the smaller and keep the remainder, then repeat with the smaller number and that remainder until the remainder is zero. The last divisor is the GCF. For 117 and 52: 117 = 2 × 52 + 13, then 52 = 4 × 13 + 0, so the GCF is 13 in two lines. This is the method the calculator runs, and it prints the chain so you can follow it.

Improper fractions and mixed numbers

An improper fraction has a numerator greater than or equal to its denominator, so its value is at least 1. To turn it into a mixed number, divide the numerator by the denominator: the quotient is the whole number and the remainder sits over the original denominator. 7/3 is 7 ÷ 3 = 2 remainder 1, so 2 1/3. 5/3 is 1 remainder 2, so 1 2/3. Going the other way, multiply the whole number by the denominator and add the numerator: 2 3/5 is (2 × 5 + 3)/5 = 13/5, and 3 1/4 is 13/4.

The order of the last two steps matters more than most explanations admit. Reduce first, then convert. 45/10 reduces to 9/2, and 9 ÷ 2 = 4 remainder 1, so the mixed number is 4 1/2. Convert before reducing and you get 4 5/10, which has the right value in the wrong form and is usually marked down. The working on this page always reduces first.

Fraction to decimal, and back again

A fraction is a division waiting to happen, so the decimal is the numerator divided by the denominator. 3/4 is 0.75 and 9/2 is 4.5. Whether the decimal stops or goes on forever can be read off before you divide: reduce the fraction, then factor the denominator. If the only primes left are 2 and 5, the decimal terminates. Any other prime, such as 3 or 7, makes it repeat. That is why 5/8 gives 0.625 exactly while 1/9 gives 0.111... and 1/7 gives 0.142857142857..., a six digit block that cycles forever. The reference table above lists the common cases, and the calculator reports which kind of decimal your answer is and which prime caused it.

To turn a decimal into a fraction, write it over 1, multiply top and bottom by 10 once for every decimal place, then simplify. 0.75 becomes 75/100, which reduces to 3/4. 0.625 becomes 625/1000, which reduces to 5/8. A decimal with a whole part, such as 2.35, becomes 2 and 35/100, which is 2 7/20. For a percentage, multiply the decimal by 100, so 3/4 is 75 percent; the answer panel shows that figure as one line. Anything more involved, such as a percentage change or finding a percent of a number, belongs on the percentage calculator, and comparing quantities part to part is a job for the ratio calculator.

Where fractions sit in the school year

If you are helping a child with homework, it helps to know what they have already been taught. In the United States Common Core standards, adding and subtracting fractions with unlike denominators is introduced in grade 5 under standard 5.NF.A.1, which states the rule a/b + c/d = (ad + bc)/bd and works 2/3 + 5/4 = 8/12 + 15/12 = 23/12. Dividing a fraction by a fraction comes a year later, in grade 6, under 6.NS.A.1, which gives (a/b) ÷ (c/d) = ad/bc. A grade 5 student who has not met the reciprocal yet will find the division steps new, and that is expected rather than a sign they have fallen behind.